You can also ask them to provide their Openfire logs.
I apologize for my previous post. After more time working with the customer I discovered that one group appeared to not be working due to all the users in that group setting themselves offline. They did so because they felt they were being micro managed by their general manager and wanted to be left alone. Whenever the alarm bell was sounded, and I restarted the service, the users put themselves back online for a while.
So the Openfire issue I described does not exist.